A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists

الأصل · EN

For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that dim F₂ Sel₂(Eᶠ/K) ≥ dim F₂ Eᶠ(K)[2] + r₂ for every quadratic twist Eᶠ of every curve E in this family, where r₂ is the number of complex places of K. This provides a counterexample to a conjecture appearing in work of Mazur and Rubin.